# Tension within a hypothetical pulley system

The following hypothetical pulley system exists: A massless rope passes over a frictionless pulley. Particles of mass M and M+m are suspended from the two differents ends of the rope. If m=0, the tension T in the pulley rope is Mg. If instead the value m increases to infinity, the value of the tension will increase, approaching a finite constant.

Using F=ma, I understand that as m increases, the force will also increase. However, I don't understand why if m approaches infinity, then the tension will approach a constant value. Why is that? My understanding is that the F=ma equation would cause tension to approach infinity.

Additionally, if the conclusion in the hypothetical pulley system is correct, is it possible to find what constant the Tension would approach? If so, what would that value be, or how would it be calculated?

## 2 Answers

Force equation: One side: T – Mg = Ma Giving: T = Mg + Ma as the mass on the other side increases, the acceleration increases to a = g (The large mass goes into free fall.) Thus the maximum T = 2Mg.

When $$M+m$$ is small, its acceleration is controlled by the force from the rope and the force from gravity. When equal, acceleration is zero.

As we increase the mass, the gravitational force grows larger and the force from the rope becomes less important. But as the mass grows, its inertia increases as well. The limit of this is that the mass falls with acceleration $$g$$, because the pull from the rope is inconsequential.

So the limiting case for the rope is that it needs to be able to accelerate mass $$M$$ at $$a=g$$. Making the massive side more massive won't increase the acceleration past that.